# The Law of Returns to Scale

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 The Laws Of Returns to Scale

 Introduction

Long run is a period during which all factors of production can vary. Long run relationship between inputs and output of a firm is explained by the Laws of returns to scale. The term returns to scale arises in the context of a firm's Production Function.In the long run production function, all factors are variable. Therefore in the long run output can be changed by changing all the factors of production.A firm's production function could exhibit different types of returns to scale in different ranges of output.Typically, there could be Increasing returns to scale,Constant returns to scale and Diminishing returns to scale. In this section we will use the isoquants to analyse the input output relationships under the condition that both the inputs (labour and Capital) are variable and their quantity is changed proportionately and simultaneously.

 Learning Objectives After reading this chapter, you are expected to learn about: {{{3}}}

 Statement of Law

"Other things being equal in the long run, as the firm increases the quantities of all factors employed, the output may rise either more than proportionately, less than proportionately or in exactly same proportion of the change in quantities of inputs.

Symbolically, the long run production function can be written as:
Two factor model: Q'x = f(L,K)
'N' factor model:
$Qx = {f(a/b,c,d,e....n,}\bar{T})$

 KEY WORDS
Returns to Scale Long run production function

 Assumptions
1. Returns are measured in physical terms. 2. All units of factors are homogeneous. 3. Techniques of production remains constant.

 Stages of Laws of Returns to Scale

1. The Increasing Returns to Scale
2. The Constant Returns to Scale
3. The Diminishing Returns to Scale

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 Explanation of Different Stages of Laws of Returns to Scale

1. The Increasing Returns to Scale: ===

There are increasing returns to scale when a given percentage increase in input leads to a greater relative percentage increase in output.

It shows that output doubles itself even before the inputs can be doubled.In the following figure that the units of labour are measured on X-axis and units of capital on Y axis. The scale line OS is drawn which shows the expansion path of a firm. In this case the distance between every successive isoquants becomes smaller and smaller i.e. OA > AB > BC.

Diagram

In case of increasing returns to scale, the production function is homogeneous of degree greater than one.

Example:
100 units (IQ1 at A) = 3L+ 3K
200 units (IQ2 at B) = 5L + 5K
300 units (IQ3 at C) = 6L + 6K

How to read example (1) : 100 units of output requires three units of labour and three units of capital.

Causes of Increasing Returns to Scale:

a.Internal economies of scale
b.Efficiency of labour and capital
c.Improvement in large scale operation
d.Division of labour and specialization
e.Use of better and sophisticated technology
f.Economy of organisation
g.External economies of scale

### 2.Constant Returns to Scale

There are constant returns to scale when a given percentage increase in input leads to an equal percentage increase in output. It shows that if inputs are doubled then the output also gets doubled. If inputs are trebled then the output also trebles

Symbolically:

Where

Proportionate Change in input

In the following figure, the units of labour are measured on X-axis and units of capital on Y-axis. OS is the scale of operation line. In this case the distance between every successive isoquant remains equal i.e. OL = LM = MN. It means if units of labour and capital are doubled, the output also doubles. In case of constant returns to scale, production function is homogenous of degree one.

Example : 100 units (IQ1 at L) = 3L + 3K 200 units (IQ2 at M) = 6L + 6K 300 units (IQ3 at N) = 9L + 9K

Diagram

Causes of Constant Returns to Scale:

a)Internal economics of scale are equal to internal diseconomies of scale.

b)Balancing of external economics and diseconomies of scale

c)Factors of production are perfectly divisible substitutable, homogenous and their supply is perfectly elastic at given prices.

### 3.Decreasing Returns to Scale

There are decreasing returns to scale when a given percentage increase in input leads to a smaller percentage increase in output.

Symbolically :

Where: Proportionate change in output.
Following figure shows the decreasing returns, where, to get an equal increase in output, a larger proportionate increase in both labour and capital are required. In case of decreasing returns to scale the distance between every successive isoquant on expansion path becomes larger and larger, i.e. OP < PQ < QR. In this case, production function is homogenous of degree less than one.

Example:

100 units (IQ, at P) = 3L + 3K 200 units (IQ2 at Q) = 7L + 7K 300 units (IQ3 at R) = 12L +12K

Diagram

Causes of Decreasing Returns to Scale

a.Internal diseconomies of scale

b.External diseconomies of scale

c.Increase in business risk

d.Lack of entrepreneurial efficiency

e.Unhealtny management and organization

f.Imperfect factor substitutability

g.Transport bottlenecks and Marketing difficulties.